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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 18496i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
18496.j4 | 18496i1 | \([0, 0, 0, 289, 0]\) | \(1728\) | \(-1544804416\) | \([2]\) | \(5120\) | \(0.45265\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
18496.j3 | 18496i2 | \([0, 0, 0, -1156, 0]\) | \(1728\) | \(98867482624\) | \([2, 2]\) | \(10240\) | \(0.79922\) | \(-4\) | |
18496.j1 | 18496i3 | \([0, 0, 0, -12716, -550256]\) | \(287496\) | \(790939860992\) | \([2]\) | \(20480\) | \(1.1458\) | \(-16\) | |
18496.j2 | 18496i4 | \([0, 0, 0, -12716, 550256]\) | \(287496\) | \(790939860992\) | \([2]\) | \(20480\) | \(1.1458\) | \(-16\) |
Rank
sage: E.rank()
The elliptic curves in class 18496i have rank \(2\).
Complex multiplication
Each elliptic curve in class 18496i has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 18496.2.a.i
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.